Exponentially weighting Bernoulli trials for a Beta prior

gompertz.R

# For the standard conjugate beta prior for a binomial likelihood, a typical
# approach is to weight each prior observation equally, there are times where
# the prior Bernoulli trials should be weighted over time, so that the more
# recent trials are weighted near 1 and the oldest trials should be weighted
# near 0.

# Gompertz Function
# http://en.wikipedia.org/wiki/Gompertz_function
gompertz <- function(x, a=1, b=1, c=1) {
  a * exp(-b * exp(-c * x))
}

week <- seq(0, 52, by=1)

# Standard Gompertz
plot(gompertz(week), type='l', main='Standard Gompertz')

# Hand-selected values to get desired weights for each week over the last year.
# Perhaps a better approach is to artifically construct a small data set of
# desired values and then to estimate the appropriate parameters.
plot(gompertz(week, 1, 20, 1/8),
     type='l',
     main="Gompertz with a=1, b=20, c=1/8")